Interpolation of Spatial Data

Transcription

1 Michael L. Stein Interpolation of Spatial Data Some Theory for Kriging With 27 Illustrations Springer

2 Contents Preface vii 1 Linear Prediction Introduction Best linear prediction 2 Exercises Hubert spaces and prediction. 4 Exercises An example of a poor BLP 6 Exercises Best linear unbiased prediction 7 Exercises Some recurring themes 10 The Matern model 12 BLPs and BLUPs 12 Inference for differentiable random nelds 13 Nested modeis are not tenable Summary of practical suggestions 14 2 Properties of Random Fields Preliminaries 15 Stationarity 16 Isotropy 17 Exercise The turning bands method 17

3 xiv Contents Exercise Elementary properties of autocovariance functions Exercise Mean Square continuity and differentiability 20 Exercises Spectral methods 22 Spectral representation of a random field 23 Bochner's Theorem 24 Exercises Two corresponding Hubert Spaces 26 An application to mean square differentiability 26 Exercises Examples of spectral densities on R 27 Rational spectral densities 28 Principal irregulär term 28 Gaussian model 29 Triangulär autocovariance functions 30 Matern class 31 Exercises Abelian and Tauberian theorems 33 Exercises Random fields with nonintegrable spectral densities Intrinsic random functions 36 Semivariograms 39 Generalized random fields 40 Exercises Isotropie autocovariance functions 42 Characterization 42 Lower bound on isotropic autocorrelation functions Inversion formula 46 Smoothness properties 46 Matern class 48 Spherical model 52 Exercises Tensor produet autocovariances 54 Exercises 55 3 Asymptotic Properties of Linear Predictors Introduction Finite sample results 59 Exercise The role of asymptotics Behavior of prediction errors in the frequency domain.. 63 Some examples 63 Relationship to filtering theory 65

4 Contents xv Exercises Prediction with the wrong spectral density 66 Examples of interpolation 66 An example with a triangulär autocovariance function.. 67 More criticism of Gaussian autocovariance functions Examples of extrapolation 70 Pseudo-BLPs with spectral densities misspecified at high frequencies 71 Exercises Theoretical comparison of extrapolation and interpolation 76 An interpolation problem 77 An extrapolation problem 78 Asymptotics for BLPs 79 Inefficiency of pseudo-blps with misspecified high frequency behavior 81 Presumed mses for pseudo-blps with misspecified high frequency behavior 85 Pseudo-BLPs with correctly specified high frequency behavior 86 Exercises Measurement errors 94 Some asymptotic theory 95 Exercises Observations on an infinite lattice 97 Characterizing the BLP 98 Bound on fraction of mse of BLP attributable to a set of frequencies 99 Asymptotic optimality of pseudo-blps 101 Rates of convergence to optimality 104 Pseudo-BLPs with a misspecified mean function 105 Exercises Equivalence of Gaussian Measures and Prediction Introduction Equivalence and orthogonality of Gaussian measures Conditions for orthogonality 111 Gaussian measures are equivalent or orthogonal 114 Determining equivalence or orthogonality for periodic ran'dom fields 118 Determining equivalence or orthogonality for nonperiodic random fields 119 Measurement errors and equivalence and orthogonality Proof of Theorem Exercises 126

5 xvi Contents 4.3 Applications of equivalence of Gaussian measures to linear prediction 129 Asymptotically optimal pseudo-blps 130 Observations not part of a sequence 132 A theorem of Blackwell and Dubins 134 Weaker conditions for asymptotic optimality of pseudo-blps 135 Rates of convergence to asymptotic optimality 138 Asymptotic optimality of BLUPs 138 Exercises Jeffreys's law 140 A Bayesian version 141 Exercises Integration of Random Fields Introduction Asymptotic properties of simple average 145 Results for sufficiently smooth random fields 147 Results for sufficiently rough random fields 148 Exercises Observations on an infinite lattice 150 Asymptotic mse of BLP 150 Asymptotic optimality of simple average 153 Exercises Improving on the sample mean 153 Approximating / 0 exp(ii/t)dt 153 Approximating /, Q x, d exp(iu; T x)dx in more than one dimension 155 Asymptotic properties of modified predictors 156 Are centered systematic samples good designs? 157 Exercises Numerical results 157 Exercises Predicting With Estimated Parameters Introduction Microergodicity and equivalence and orthogonality of Gaussian measures 162 Observations with measurement error 164 Exercises Is Statistical inference for differentiable processes possible? 166 An example where it is possible 167 Exercises 168

6 Contents xvii 6.4 Likelihood Methods 169 Restricted maximum likelihood estimation 170 Gaussian assumption 171 Computational issues 172 Some asymptotic theory 174 Exercises Matern model 176 Exercise A numerical study of the Fisher information matrix under the Matern model 178 No measurement error and v unknown 179 No measurement error and v known 181 Observations with measurement error 182 Conclusions 186 Exercises Maximum likelihood estimation for a periodic Version of the Matern model 188 Discrete Fourier transforms 188 Periodic case 189 Asymptotic results 190 Exercises 198' 6.8 Predicting with estimated parameters 199 Jeffreys's law revisited 203 Numerical results 206 Some issues regarding asymptotic optimality 210 Exercises An instructive example of plug-in prediction 211 Behavior of plug-in predictions 214 Cross-validation 215 Application of Matern model 218 Conclusions 220 Exercises Bayesian approach 223 Application to simulated data 225 Exercises 226 A Multivariate Normal Distributions 229 B Symbols 231 References 235 Index 243